, 2012

The average of any two numbers is found by adding the two numbers and dividing by two.

Thus, the three averages when the numbers are taken in pairs are $$\text{25+5 2 = 30 2 =15, , , 5+29 2 = 34 2 =17, and 25+29 2 = 54 2 =27.}$$
The average of 2 and 6 is $\dfrac{2+6}{2}=4$.

Since 6 is greater than 2, then the average of 6 and $n$ is greater than the average of 2 and $n$.

Therefore, the average of 6 and $n$ is 13 and the average of 2 and $n$ is 11.

Since the average of 6 and $n$ is 13, then $\dfrac{6+n}{2}=13$ or $6+n=26$ and so $n=20$.

We can check that $n=20$ is correct by recognizing that the average of 2 and 20 is indeed 11.
When each of the three numbers is added to the average of the other two, the resulting three expressions are $$2+\frac{a+b}{2},\,\,a+\frac{2+b}{2},\,\,b+\frac{2+a}{2}.$$ To determine which of these expressions is equal to which of the results, 14, 17, 21, we must order the three expressions from smallest to largest.

Since $\text{2 lt;a lt;b}$, then $\text{2+(2+a+b) lt;a+(2+a+b) lt;b+(2+a+b)}$

or $\text{4+a+b lt;2a+2+b lt;2b+2+a}$.

Dividing by 2, $\text{4+a+b 2 lt; 2a+2+b 2 lt; 2b+2+a 2}$ or $\text{4 2 + a+b 2 lt; 2a 2 + 2+b 2 lt; 2b 2 + 2+a 2}$

and so $\text{2+ a+b 2 lt;a+ 2+b 2 lt;b+ 2+a 2}$.

Since $2+\dfrac{a+b}{2}$ is the smallest of the three expressions, then it must equal the smallest of the three results, 14.

Since $b+\dfrac{2+a}{2}$ is the largest of the three expressions, then it must equal the largest of the three results, 21.

We now solve the following system of two equations and two unknowns. $$\begin{align*} 2+\dfrac{a+b}{2} & = 14 \tag{1} \\ b+\dfrac{2+a}{2} & = 21 \tag{2} \end{align*}$$ Multiplying each equation by 2, $$\begin{align*} 4+a+b & = 28 \tag{3} \\ 2b+2+a & = 42 \tag{4} \end{align*}$$ Thus, $$\begin{align*} a+b & = 24 \tag{5}\\ a+2b & = 40 \tag{6} \end{align*}$$ Subtracting equation $(5)$ from equation $(6)$, we get $b=16$.

Substituting $b=16$ into equation $(5)$, $a+16=24$, and so $a=8$.

(We may check that our solution is correct by substituting $a=8$ and $b=16$ into the third expression $a+\dfrac{2+b}{2}$ to get the third result, 17.)